What am I doing wrong to calculate this standard deviation?

Question

This post mentions in passing that the standard deviation of the data set (1, 50) is about 34.65. Wolfram Alpha confirms this, saying it is $\frac{49}{\sqrt{2}}\approx 34.648$. But when I try to calculate this by hand, I get:

$\mu = \frac{1+50}{2}=\frac{51}{2}=25.5$

$\sigma = \sqrt{\frac{(25.5-1)^2+(25.5-50)^2}{2}}=\sqrt{\frac{(24.5)^2+(-24.5)^2}{2}}=\sqrt{\frac{2\cdot(24.5)^2}{2}}=24.5$.

What am I doing wrong?

Answer 1

You actually did standard deviation correct, what you did is called the population standard deviation.

What the original problem was talking about and wolfram alpha said is called sample standard deviation.

Sample standard deviation is calculated with this formula: $s= \sqrt{\frac{\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}{N-1}}$

Answer 2

If $(1,50)$ represents the population, you divide by $n=2$ in the calculation of standard deviation, but if it's a sample from the population (in this case, of size $2$), then you divide by $n-1=2-1$ in this case. The former refers to the population standard deviation and the latter the sample standard deviation.

$$s=\sqrt{\frac{2\cdot (24.5)^2}{2-1}}=34.65$$